I don't see what I should use here. What would you use?
$$\frac{2^{2013}}{2013}\le\frac{\binom{2013}{0}}{1}+\frac{\binom{2013}{1}}{3}+\frac{\binom{2013}{2}}{5}+\cdots+\frac{\binom{2013}{2013}}{2\cdot 2013+1}\le\frac{2^{2013}}{2012}$$